The French research system : which evolution and which borders ?

We analyse the French Research System with the study of the contracts between the CNRS (Centre National de la Recherche Scientifique) and the companies and test the hypothesis of small world in science. Our working material is the data base of the contracts of the units of the CNRS with economic partners, which has been collecting information since 1986 to 2006. This first application of Network methods and tools to the CNRS contracts allows us to obtain some results: at first, the major firms’s scientific network is not "scale-free" as if competition and strategy between the most large firms dominate the behaviour in R&D investments and management of contracts with public research units. However, in second part, we demonstrate that every discipline network is a "small world", i.e. , that it exists several scientific communities in which the diffusion of information is free and easy, even if its forwards through any actors (some labs or some firms). Probably, there are several "small worlds" in this database as in the scientific collaboration networks. Is seems that the industrial research does not disturb too much the properties of scientific network, as it’s well known in the literature of Sciences Studies

Large deviations and continuity estimates for the derivative of a random model of log⁡∣ζ∣\log |\zeta| on the critical line

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for the large values of $(\log |\zeta(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin, Belius & Harper (2017) up to the second order, but the tightness of the recentered maximum is still an open problem. As a first step, we provide large deviation estimates and continuity estimates for the field's derivative $X'(h)$. The main result shows that, with probability arbitrarily close to $1$, \begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1), \end{equation*} where $\mathcal{S}$ a discrete set containing $O(\log T \sqrt{\log \log T})$ points.Comment: 7 pages, 0 figur

Quantum entropic security and approximate quantum encryption

We present full generalisations of entropic security and entropic indistinguishability to the quantum world where no assumption but a limit on the knowledge of the adversary is made. This limit is quantified using the quantum conditional min-entropy as introduced by Renato Renner. A proof of the equivalence between the two security definitions is presented. We also provide proofs of security for two different cyphers in this model and a proof for a lower bound on the key length required by any such cypher. These cyphers generalise existing schemes for approximate quantum encryption to the entropic security model.Comment: Corrected mistakes in the proofs of Theorems 3 and 6; results unchanged. To appear in IEEE Transactions on Information Theory

Extremes of the two-dimensional Gaussian free field with scale-dependent variance

In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.Comment: 30 pages, 4 figures. The argument in Lemma 3.1 and 3.4 was revised. Lemma A.4, A.5 and A.6 were added for this reason. Other typos were corrected throughout the article. The proof of Lemma A.1 and A.3 was simplifie

Building Decision Procedures in the Calculus of Inductive Constructions

It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P' obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proof-checking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved

Variational Monte-Carlo investigation of SU(NN) Heisenberg chains

Motivated by recent experimental progress in the context of ultra-cold multi-color fermionic atoms in optical lattices, we have investigated the properties of the SU($N$) Heisenberg chain with totally antisymmetric irreducible representations, the effective model of Mott phases with $m < N$ particles per site. These models have been studied for arbitrary $N$ and $m$ with non-abelian bosonization [I. Affleck, Nuclear Physics B 265, 409 (1986); 305, 582 (1988)], leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. Using exact diagonalization and variational Monte-Carlo based on Gutzwiller projected fermionic wave functions, we have been able to verify these predictions for a representative number of cases with $N \leq 10$ and $m \leq N/2$, and we have shown that the opening of a gap is associated to a spontaneous dimerization or trimerization depending on the value of m and N. We have also investigated the marginal cases where abelian bosonization did not lead to any prediction. In these cases, variational Monte-Carlo predicts that the ground state is critical with exponents consistent with conformal field theory.Comment: 9 pages, 10 figures, 3 table

A uniform L1L^1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator

We develop a new $L^1$ law of large numbers where the $i$-th summand is given by a function $h(\cdot)$ evaluated at $X_i - \theta_n$, and where $\theta_n \circeq \theta_n(X_1,X_2,\ldots,X_n)$ is an estimator converging in probability to some parameter $\theta\in \mathbb{R}$. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between $\theta$ and another consistent estimator $\theta_n^{\star}$. Our main contribution is the treatment of the case where $|h|$ blows up at $0$, which is not covered by standard uniform laws of large numbers.Comment: 10 pages, 1 figur